| L(s) = 1 | − 2·3-s + 3·9-s − 4·17-s − 8·23-s − 6·25-s − 4·27-s − 12·29-s − 8·43-s + 14·49-s + 8·51-s − 4·53-s + 20·61-s + 16·69-s + 12·75-s − 16·79-s + 5·81-s + 24·87-s + 12·101-s + 16·103-s − 36·113-s + 18·121-s + 127-s + 16·129-s + 131-s + 137-s + 139-s − 28·147-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s − 0.970·17-s − 1.66·23-s − 6/5·25-s − 0.769·27-s − 2.22·29-s − 1.21·43-s + 2·49-s + 1.12·51-s − 0.549·53-s + 2.56·61-s + 1.92·69-s + 1.38·75-s − 1.80·79-s + 5/9·81-s + 2.57·87-s + 1.19·101-s + 1.57·103-s − 3.38·113-s + 1.63·121-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.30·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16451136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16451136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1385756406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1385756406\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 138 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.997288635621098660284911847112, −8.226660941077146572274547627181, −7.82437074055018300681173828612, −7.39494316028735243940451077596, −7.18914116641740289640708927112, −6.74724172232196270335050684510, −6.37311257314733053783950046564, −5.93812720205782917887906178064, −5.58206465061934951192235524288, −5.56173143830816768412169856756, −4.88930915865677718139200468288, −4.48536748280519740277606260594, −4.04026961310307234749478073424, −3.77910002239430858907043900061, −3.42141294039053688788336290104, −2.49689490673139884913913650682, −2.05941343744698439545371308178, −1.82773280301130631086205581584, −1.02532474988957540364259615939, −0.12714425726434420776516908787,
0.12714425726434420776516908787, 1.02532474988957540364259615939, 1.82773280301130631086205581584, 2.05941343744698439545371308178, 2.49689490673139884913913650682, 3.42141294039053688788336290104, 3.77910002239430858907043900061, 4.04026961310307234749478073424, 4.48536748280519740277606260594, 4.88930915865677718139200468288, 5.56173143830816768412169856756, 5.58206465061934951192235524288, 5.93812720205782917887906178064, 6.37311257314733053783950046564, 6.74724172232196270335050684510, 7.18914116641740289640708927112, 7.39494316028735243940451077596, 7.82437074055018300681173828612, 8.226660941077146572274547627181, 8.997288635621098660284911847112
